46 M. Braun
✁
1 2
✂ ✄
3 4
☎ ✆
5 6
✝ ✞
✟ ✠
✡ ☛
2n − 2
☞ ✌
2n 2n
− 1
✍ ✎
✏ ✑
Figure 13: Regular rosette
✒ ✓
✒ ✓
✒ ✓
✒ ✓
✔ ✕
1 3
5 7
2
4
6 8
Figure 14: Quadratic rosette
This is the analogue of S
T
.-V
ENANT
’s compatibility condition for a truss. It is obtained by combining the closing condition 13 for the rotations around a rosette with the closing condition
12 for the displacements around a triangle. For the truss to be stress-free in its unloaded state the condition 14 is necessary but not sufficient, in general. If the 2-complex does not contain
any holes, the condition is also sufficient. The compatibility conditions are closely connected with the extended model of the truss as an oriented 2-complex, although the 2-simplexes are not
material parts of the truss.
In the special case of a regular rosette Figure 13 all the angles α
i,i +1
are equal and cancel out. Thus the compatibility condition reduces to
n
X
i =1
ε
2i −1
=
n
X
i =1
ε
2i
. The sum of the circumferential strains must be equal to the sum of the radial strains. The general
compatibility condition 14 has a similar structure, with the strains being affected by certain geometrical weight factors. For a quadratic rosette the compatibility condition reads
ε
1
+ ε
3
+ ε
5
+ ε
7
= ε
2
+ ε
4
+ ε
6
+ ε
8
. This equation can be interpreted as a discretization of S
T
.-V
ENANT
’s compatibility condition 9.
7. Conclusion
The general structure of elasticity theory is not confined to the continuum version, but holds also for discrete elastic systems such as trusses or finite-element models. A remarkable difference be-
tween the theories of plane trusses and of elastic continua is the fact that in the continuous case all quantities are declared throughout the whole body, whereas in the discrete case of the truss
each quantity has its own “carrier”: Displacements are declared in the nodes, strain and rota- tion are available in the members, rotation differences need pairs of members, and compatibility
conditions can be formulated for “rosettes”, ı. e. inner nodes that are completely surrounded by triangles of truss members. In this sense the continuum theory could be regarded as “easier”,
Compatibility conditions 47
since all quantities are defined in each material point. A closer look shows, however, that the continuum theory can also provide different carriers for different quantities. This becomes mani-
fest, if the mechanical quantities are described in terms of differential forms rather than ordinary field functions.
∗
The compatibility condition for a truss have been developed using the same ideas as in the continuum. It rests upon the postulation that displacement and rotation can be represented by
path-independent integrals or, in the discrete case, by path-independet finite sums. To generate localized integrability conditions in a continuum the integral around a closed path is transformed
via S
TOKES
’s theorem into a surface integral, which must vanish identically. In the truss case the local conditions are obtained by choosing the smallest nontrivial closed paths or 1-cycles,
namely triangles for the displacements and rosettes for the rotations. The theory of trusses can be developed further and extended along these lines. The com-
patibility condition should be complemented by its dual, the representation of member forces by A
IRY
’s stress function. This quantity has the same carrier as the compatibility condition, i. e., it resides in the rosettes surrounding inner nodes of the truss. The generalization to three
dimensions is more intricate, especially with respect to the closing condition for the rotation vector.
Quite interesting is the appropriate treatment of frame trusses, with members rigidly clamped to each other. A frame truss allows forces and couples to be applied to the nodes, and its mem-
bers deform under extension, bending, and torsion. In this case the corresponding continuum theory has to include couple stresses. It might be interesting to compare the common features of
continuous and discrete couple-stress theories.
Also a nonlinear theory of trusses can be formulated from the paradigm of nonlinear elas- ticity theory. The concept of different placements is easily transferred to a truss, and also the
E
SHELBY
stress tensor has its counterpart in the discrete case. A first attempt in this direction has been made by the author in [2].
References
[1] A
NTHONY
K. H., Die Theorie der Disklinationen Archive for Rational Mechanics and
Analysis 39 1970, 43–88.
[2] B
RAUN
M., Continuous and discrete elastic structures, Proceedings of the Estonian
Academy of Sciences, Physics–Mathematics 48 1999, 174–188.
[3] C
ROOM
F. H. Basic concepts of algebraic topology, Springer Verlag, New York Heidelberg Berlin 1978.
[4] K
LEIN
F.
AND
W
IEGHARDT
K. ¨ Uber Spannungsfl¨achen und reziproke Diagramme, mit
besonderer Ber¨ucksichtigung der Maxwellschen Arbeiten, Archiv der Mathematik und Physik III Reihe 8 1905, 1–10, 95–119.
[5] R
IEDER
G. Mechanik Festigkeitslehre, Vorlesung an der RWTH Aachen 1978. [6] T
ONTI
E. On the mathematical structure of a large class of physical theories, Atti della
Academia Nazionale dei Lincei 52 1972, 48–56.
[7] T
ONTI
E. The reason for analogies between physical theories, Applied Mathematics Mod-
elling 1 1976, 37–53 .
∗
This aspect has been pointed out by Professor A
NTONIO DI
C
ARLO
in the discussion of the paper at the Torino seminar.
48 M. Braun
AMS Subject Classification: 74B99, 55U10.
Manfred BRAUN Gerhard-Mercator-Universit¨at Duisburg,
Institute of Mechatronics and System Dynamics 47048 Duisburg, GERMANY
e-mail: braunmechanik.uni-duisburg.de
Rend. Sem. Mat. Univ. Pol. Torino Vol. 58, 1 2000
Geom., Cont. and Micros., I
M. Brocato - G. Capriz